respectively, where \(\cup _{f}\)  means gluing the boundaries
together by the diffeomorphism \(f\) . Alternatively, the manifolds \(\Sigma _{0}N\)  and \(\Sigma _{1}N\)  are obtained by surgeries along the oriented
circle \(S^{1}\times \lbrace x_{0}\rbrace \)  on \(S^{1}\times N\)  with the framings \(id\)  and
\(\widetilde{\alpha }\) , respectively. By the homogeneity lemma [1]} the diffeomorphism types of both \(\Sigma _{i}N\)  are irrelevant to the
choice of a smooth embedding \((D^{n},0)\subset (N,x_{0})\) , which justifies
the following notion.
